A Critical Monotonicity Principle A modular identity–rigidity package for constrained gradient dynamics

We formalize a Critical Monotonicity Principle (CMP) for normalized/constrained gradient dynamics in a Hilbert/Dirichlet framework. The core mechanism is a structural defect identity whose induced criticality observable is monotone and whose equality case enforces rigidity (selection of a critical state / eigenstate). We present a localized version with cutoffs and leakage, and a quantitative threshold criterion turning monotonicity into a usable selection mechanism. As a model application, we connect CMP to Rayleigh-type normalized flows generated by a self-adjoint dissipation operator and a constraint functional, clarifying how energy decay, rigidity, and localization emerge from the same identity.